The pollution effect for FEM approximations of the Ginzburg-Landau equation

TL;DR

The paper addresses the pollution effect in finite element approximations of the Ginzburg--Landau equation by deriving a fully κ-explicit error framework. It develops a shift-based analysis using $E''(u)$, a κ-weighted Ritz projection, and a Newton-type fixed-point argument to quantify pre-asymptotic error regimes, showing that higher-order finite elements (larger $p$) relax the mesh-resolution requirements needed to achieve quasi-optimal convergence in the $H^1_ abla$ norm and, under certain conditions, in the $L^2$ norm as well. The main contributions include a κ-dependent stability theory for minimizers, a precise resolution condition $ ho(oldsymbol{ abla})(holdsymbol{ abla})^p\, ext{scaled with }oldsymbol{ abla}^{d/2}$ ensuring quasi-optimality, and numerical experiments confirming reduced pollution with higher $p$. The results provide practical guidance for mesh design in superconductivity simulations, linking vortex resolution to polynomial degree and mesh size while delivering improved $H^1_ abla$ and $L^2$ error bounds.

Abstract

In this paper, we investigate the approximation properties of solutions to the Ginzburg-Landau equation (GLE) in finite element spaces. Special attention is given to how the errors are influenced by coupling the mesh size $h$ and the polynomial degree $p$ of the finite element space to the size of the so-called Ginzburg-Landau material parameter $κ$. As observed in previous works, the finite element approximations to the GLE are suffering from a numerical pollution effect, that is, the best-approximation error in the finite element space converges under mild coupling conditions between $h$ and $κ$, whereas the actual finite element solutions possess poor accuracy in a large pre-asymptotic regime which depends on $κ$. In this paper, we provide a new error analysis that allows us to quantify the pre-asymptotic regime and the corresponding pollution effect in terms of explicit resolution conditions. In particular, we are able to prove that higher polynomial degrees reduce the pollution effect, i.e., the accuracy of the best-approximation is achieved under relaxed conditions for the mesh size. We provide both error estimates in the $H^1$- and the $L^2$-norm and we illustrate our findings with numerical examples.

Figures (4)

• Figure 1: Vortex patterns of the reference solution $|u_{{\text{\tiny ref}}}|$ for $\kappa=8,16,24,32,40$ (from the upper left to the lower right picture).
• Figure 2: Error in energy $|E(u)-E(u_h)|$ for $\kappa = 8,16,24,32,40$ between the reference minimal energy $E(u)$ and the minimal energy $E(u_h)$ in the finite element spaces for P1 (solid lines) and P2 (dashed lines) respectively.
• Figure 3: Comparison of the errors $\| u - u_{h} \|_{L^2(\Omega)}$ and $\tfrac{1}{\kappa}\| \nabla u - \nabla u_{h} \|_{L^2(\Omega)}$ for the discrete minimizers $u_h$ in the FEM spaces $V_{h,1}$ (P1, solid lines) and $V_{h,2}$ (P2, dashed lines), where $\kappa = 8,16,24,32,40$.
• Figure 4: Comparison of the errors $\| u - P_h(u) \|_{L^2(\Omega)}$ and $\tfrac{1}{\kappa}\| \nabla u - \nabla P_h(u) \|_{L^2(\Omega)}$ for the $H^1_\kappa$-best-approximation $P_h(u)$ of $u$ given by \ref{['def-Phu-projec']} for $V_{h,1}$ (P1, solid lines) and $V_{h,2}$ (P2, dashed lines). The errors include the cases $\kappa = 8,16,24,32,40$.

Theorems & Definitions (64)

• Remark 2.1: Piecewise smooth coefficient
• Remark 2.2: Physically relevant states
• Lemma 2.3
• Remark 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• proof
• Proposition 3.1: Error in the minimal energy
• proof